A086500 Group the natural numbers such that the n-th group sum is divisible by the n-th triangular number: (1), (2, 3, 4), (5, 6, 7), (8, 9, 10, 11, 12), (13, 14, 15, 16, 17), (18, 19, 20, 21, 22, 23, 24), ... Sequence contains the group sum.

1, 9, 18, 50, 75, 147, 196, 324, 405, 605, 726, 1014, 1183, 1575, 1800, 2312, 2601, 3249, 3610, 4410, 4851, 5819, 6348, 7500, 8125, 9477, 10206, 11774, 12615, 14415, 15376, 17424, 18513, 20825, 22050, 24642, 26011, 28899, 30420, 33620, 35301

Offset: 1

Author

Amarnath Murthy, Jul 28 2003

Keywords

• nonn
• easy

Comments

The number of terms in the groups is given by A063196. i.e., 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, ...

Also the arithmetic mean of the n-th group is T(n), the n-th triangular number.

Links

• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Crossrefs

Cf. A001082, A022998, A063196, A181900.

Programs

Haskell

a086500 n = a086500_list !! (n-1)
a086500_list = scanl1 (+) $ tail a181900_list
-- Reinhard Zumkeller, Mar 31 2012

Mathematica

Table[n*(n + 1)*(2*n + 1 + (-1)^n)/4, {n, 1, 40}] (* Amiram Eldar, Feb 22 2022 *)

PARI

Vec(x*(x^4+8*x^3+6*x^2+8*x+1)/((x-1)^4*(x+1)^3) + O(x^100)) \\ Colin Barker, Sep 20 2014

Formula

a(n) = n*(n+1)*(2*n+1+(-1)^n)/4. - Wesley Ivan Hurt, Sep 19 2014

a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7) for n>7. - Colin Barker, Sep 19 2014

G.f.: x*(x^4+8*x^3+6*x^2+8*x+1) / ((x-1)^4*(x+1)^3). - Colin Barker, Sep 19 2014

From Amiram Eldar, Feb 22 2022: (Start)

Sum_{n>=1} 1/a(n) = 4*(1-log(2)).

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/2 - 4. (End)

Extensions

More terms from Ray Chandler, Sep 17 2003